← #12068218
Subgroups — #12068218
Order 1280.
900 conjugacy classes of subgroups.
The subgroup lattice has 296 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10617729 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×#2991 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10614254 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10601571 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10601570 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10601304 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #10601301 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C4×C5×#41 | C22 | 1 | 1 | yes | characteristic |
| 9 | #66161 | C4 | 1 | 2 | yes | — |
| 10 | #65904 | — | 2 | 1 | no | — |
| 11 | Q8×Dic10 | — | 2 | 1 | no | — |
| 12 | #65877 | — | 2 | 3 | no | — |
| 13 | D4×Dic10 | — | 2 | 3 | no | — |
| 14 | #65853 | C4 | 1 | 6 | yes | — |
| 15 | #65804 | C22 | 1 | 3 | yes | — |
| 16 | C2×#4015 | C22 | 1 | 3 | yes | — |
| 17 | C5×C8×Q8 | — | 2 | 1 | no | — |
| 18 | C5×#436 | — | 2 | 3 | no | — |
| 19 | C5×#431 | C22 | 1 | 3 | yes | — |
| 20 | C5×#423 | C22 | 1 | 3 | yes | — |
| 21 | Q8×#168 | — | 2 | 1 | no | — |
| 22 | #65301 | — | 2 | 3 | no | — |
| 23 | #65214 | C22 | 1 | 3 | yes | — |
| 24 | #65213 | C22 | 1 | 3 | yes | — |
| 25 | #64706 | C22 | 1 | 3 | yes | — |
| 26 | #64706 | C22 | 1 | 6 | yes | — |
| 27 | #64706 | C22 | 1 | 3 | yes | — |
| 28 | #64702 | C22 | 1 | 3 | yes | — |
| 29 | #64702 | C22 | 1 | 1 | yes | characteristic |
| 30 | #17777 | — | 5 | 1 | no | maximal, Sylow |
| 31 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C22×Dic10 | — | 2 | 3 | no | — |
| 33 | C2×C5×M16 | — | 2 | 3 | no | — |
| 34 | C4×C5×Q8 | D4 | 1 | 1 | yes | characteristic |
| 35 | C4×C5×Q8 | D4 | 1 | 1 | yes | characteristic |
| 36 | C4×C5×D4 | D4 | 1 | 3 | yes | — |
| 37 | C4×C5×D4 | D4 | 1 | 3 | yes | — |
| 38 | C5×#117 | C2×C4 | 1 | 3 | yes | — |
| 39 | C2×C42×C5 | C23 | 1 | 3 | yes | — |
| 40 | Q8×Dic5 | — | 2 | 2 | no | — |
| 41 | D4×Dic5 | — | 2 | 6 | no | — |
| 42 | #4072 | C2×C4 | 1 | 6 | yes | — |
| 43 | C2×C4⋊Dic5 | — | 2 | 3 | no | — |
| 44 | C2×C4⋊Dic5 | C2×C4 | 1 | 6 | yes | — |
| 45 | #4070 | — | 2 | 3 | no | — |
| 46 | #4070 | — | 2 | 3 | no | — |
| 47 | C2×#681 | — | 2 | 3 | no | — |
| 48 | #4036 | — | 2 | 6 | no | — |
| 49 | #4034 | — | 2 | 6 | no | — |
| 50 | #4024 | — | 2 | 12 | no | — |
| 51 | #4016 | C2×C4 | 1 | 6 | yes | — |
| 52 | #4015 | — | 2 | 3 | no | — |
| 53 | #4015 | — | 2 | 3 | no | — |
| 54 | #4015 | C2×C4 | 1 | 6 | yes | — |
| 55 | #4015 | C23 | 1 | 3 | yes | — |
| 56 | #4015 | C23 | 1 | 1 | yes | characteristic |
| 57 | C4×Dic10 | — | 2 | 3 | no | — |
| 58 | C4×Dic10 | — | 2 | 3 | no | — |
| 59 | C5×C4⋊C8 | — | 2 | 3 | no | — |
| 60 | C5×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 61 | C5×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 62 | C5×C22⋊C8 | — | 2 | 3 | no | — |
| 63 | C4×C5×C8 | — | 2 | 3 | no | — |
| 64 | C22⋊#168 | — | 2 | 3 | no | — |
| 65 | C4⋊#168 | — | 2 | 3 | no | — |
| 66 | C4⋊#168 | C23 | 1 | 3 | yes | — |
| 67 | C4⋊#168 | C23 | 1 | 1 | yes | characteristic |
| 68 | C4×#168 | — | 2 | 3 | no | — |
| 69 | #3503 | — | 5 | 1 | no | — |
| 70 | #2991 | — | 5 | 1 | no | — |
| 71 | #2991 | D5 | 1 | 1 | yes | characteristic |
| 72 | #1561 | — | 5 | 3 | no | — |
| 73 | #1549 | — | 5 | 3 | no | — |
| 74 | #1546 | — | 5 | 3 | no | — |
| 75 | #1544 | — | 5 | 3 | no | — |
| 76 | C5×#41 | — | 2 | 4 | no | — |
| 77 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 78 | C2×C5×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 79 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 80 | C22×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 81 | C22×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 82 | C22×Dic5 | — | 2 | 6 | no | — |
| 83 | C2×Dic10 | — | 2 | 3 | no | — |
| 84 | C2×Dic10 | — | 2 | 1 | no | — |
| 85 | C2×Dic10 | — | 2 | 6 | no | — |
| 86 | C2×Dic10 | — | 8 | 3 | no | — |
| 87 | C2×Dic10 | — | 8 | 3 | no | — |
| 88 | C5×M16 | — | 8 | 3 | no | — |
| 89 | C2×C5×C8 | — | 2 | 3 | no | — |
| 90 | C2×C5×C8 | — | 2 | 1 | no | — |
| 91 | C5×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 92 | C5×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 93 | C5×C22⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 94 | C42×C5 | C22×C4 | 1 | 3 | yes | — |
| 95 | C42×C5 | C2×D4 | 1 | 3 | yes | — |
| 96 | C42×C5 | C2×D4 | 1 | 3 | yes | — |
| 97 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 98 | C22⋊Dic5 | — | 2 | 6 | no | — |
| 99 | C4⋊Dic5 | — | 2 | 6 | no | — |
| 100 | C4⋊Dic5 | — | 2 | 6 | no | — |
| 101 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 102 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 103 | C4⋊Dic5 | C22×C4 | 1 | 6 | yes | — |
| 104 | C4⋊Dic5 | C22×C4 | 1 | 2 | yes | — |
| 105 | #683 | — | 2 | 6 | no | — |
| 106 | #683 | — | 2 | 6 | no | — |
| 107 | C4×Dic5 | — | 2 | 6 | no | — |
| 108 | #681 | — | 8 | 3 | no | — |
| 109 | C2×#168 | — | 2 | 1 | no | — |
| 110 | C2×#168 | — | 2 | 3 | no | — |
| 111 | Q82 | — | 10 | 1 | no | — |
| 112 | #557 | — | 10 | 1 | no | — |
| 113 | #554 | — | 10 | 3 | no | — |
| 114 | D4×Q8 | — | 10 | 3 | no | — |
| 115 | #544 | — | 5 | 6 | no | — |
| 116 | #533 | — | 5 | 3 | no | — |
| 117 | C2×C4⋊Q8 | — | 5 | 3 | no | — |
| 118 | #520 | — | 5 | 2 | no | — |
| 119 | C4×#41 | C2×D5 | 1 | 1 | yes | characteristic |
| 120 | C8×Q8 | — | 2 | 1 | no | — |
| 121 | C8×Q8 | — | 10 | 1 | no | — |
| 122 | #436 | — | 2 | 3 | no | — |
| 123 | #436 | — | 10 | 3 | no | — |
| 124 | #431 | — | 5 | 3 | no | — |
| 125 | #431 | C2×D5 | 1 | 3 | yes | — |
| 126 | #423 | — | 5 | 3 | no | — |
| 127 | #423 | C2×D5 | 1 | 3 | yes | — |
| 128 | #332 | — | 5 | 1 | no | — |
| 129 | #332 | — | 5 | 3 | no | — |
| 130 | #332 | — | 5 | 3 | no | — |
| 131 | #332 | — | 5 | 3 | no | — |
| 132 | #332 | — | 5 | 6 | no | — |
| 133 | C23×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 134 | C5×Q8 | #101 | 1 | 4 | yes | — |
| 135 | C5×D4 | — | 2 | 6 | no | — |
| 136 | C2×C4×C5 | — | 2 | 3 | no | — |
| 137 | C2×C4×C5 | — | 2 | 3 | no | — |
| 138 | C2×C4×C5 | — | 2 | 3 | no | — |
| 139 | C2×C4×C5 | — | 2 | 3 | no | — |
| 140 | C2×C4×C5 | — | 2 | 6 | no | — |
| 141 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 142 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 143 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 144 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 147 | C2×Dic5 | — | 2 | 2 | no | — |
| 148 | C2×Dic5 | — | 2 | 6 | no | — |
| 149 | C2×Dic5 | — | 8 | 6 | no | — |
| 150 | Dic10 | — | 4 | 3 | no | — |
| 151 | Dic10 | — | 4 | 1 | no | — |
| 152 | Dic10 | — | 4 | 3 | no | — |
| 153 | Dic10 | — | 4 | 1 | no | — |
| 154 | Dic10 | — | 8 | 6 | no | — |
| 155 | C5×C8 | — | 4 | 3 | no | — |
| 156 | C5×C8 | — | 4 | 1 | no | — |
| 157 | #168 | — | 4 | 3 | no | — |
| 158 | #168 | — | 4 | 1 | no | — |
| 159 | C2×#41 | C4×D5 | 1 | 1 | yes | characteristic |
| 160 | C22×Q8 | — | 10 | 3 | no | — |
| 161 | C2×M16 | — | 2 | 3 | no | — |
| 162 | C2×M16 | — | 10 | 3 | no | — |
| 163 | C4⋊Q8 | — | 5 | 6 | no | — |
| 164 | C4⋊Q8 | — | 5 | 3 | no | — |
| 165 | C4⋊Q8 | — | 5 | 1 | no | — |
| 166 | C4⋊Q8 | — | 10 | 6 | no | — |
| 167 | C4⋊Q8 | — | 10 | 3 | no | — |
| 168 | C4⋊Q8 | — | 10 | 3 | no | — |
| 169 | #125 | — | 5 | 6 | no | — |
| 170 | #125 | — | 10 | 6 | no | — |
| 171 | C22⋊Q8 | — | 10 | 3 | no | — |
| 172 | C22⋊Q8 | — | 10 | 3 | no | — |
| 173 | C22⋊Q8 | — | 10 | 12 | no | — |
| 174 | C4×Q8 | — | 10 | 2 | no | — |
| 175 | C4×Q8 | — | 10 | 3 | no | — |
| 176 | C4×Q8 | — | 10 | 3 | no | — |
| 177 | C4×Q8 | D20 | 1 | 1 | yes | characteristic |
| 178 | C4×Q8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 179 | C4×D4 | — | 10 | 6 | no | — |
| 180 | C4×D4 | D20 | 1 | 3 | yes | — |
| 181 | C4×D4 | C22⋊D5 | 1 | 3 | yes | — |
| 182 | #117 | — | 5 | 6 | no | — |
| 183 | #117 | C4×D5 | 1 | 3 | yes | — |
| 184 | C2×C4⋊C4 | — | 5 | 6 | no | — |
| 185 | C2×C4⋊C4 | — | 10 | 3 | no | — |
| 186 | C2×C42 | C22×D5 | 1 | 3 | yes | — |
| 187 | C4⋊C8 | — | 2 | 3 | no | — |
| 188 | C4⋊C8 | — | 5 | 3 | no | — |
| 189 | C4⋊C8 | — | 5 | 1 | no | — |
| 190 | C4⋊C8 | — | 10 | 3 | no | — |
| 191 | C4⋊C8 | C22×D5 | 1 | 3 | yes | — |
| 192 | C4⋊C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 193 | C22⋊C8 | — | 2 | 3 | no | — |
| 194 | C22⋊C8 | — | 10 | 3 | no | — |
| 195 | C4×C8 | — | 2 | 3 | no | — |
| 196 | C4×C8 | — | 10 | 3 | no | — |
| 197 | C22×C5 | — | 2 | 3 | no | — |
| 198 | C22×C5 | — | 2 | 3 | no | — |
| 199 | C22×C5 | — | 2 | 3 | no | — |
| 200 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 201 | C4×C5 | — | 2 | 1 | no | — |
| 202 | C4×C5 | — | 2 | 3 | no | — |
| 203 | C4×C5 | — | 2 | 3 | no | — |
| 204 | C4×C5 | C2×#101 | 1 | 6 | yes | — |
| 205 | C4×C5 | #576 | 1 | 1 | yes | characteristic |
| 206 | C4×C5 | #577 | 1 | 1 | yes | characteristic |
| 207 | C4×C5 | #577 | 1 | 1 | yes | characteristic |
| 208 | C4×C5 | #578 | 1 | 1 | yes | characteristic |
| 209 | Dic5 | — | 4 | 6 | no | — |
| 210 | Dic5 | — | 4 | 2 | no | — |
| 211 | #41 | — | 2 | 4 | no | — |
| 212 | C2×Q8 | — | 10 | 3 | no | — |
| 213 | C2×Q8 | — | 10 | 1 | no | — |
| 214 | C2×Q8 | — | 10 | 6 | no | — |
| 215 | C2×Q8 | — | 40 | 3 | no | — |
| 216 | C2×Q8 | — | 40 | 3 | no | — |
| 217 | C2×Q8 | #685 | 1 | 1 | yes | characteristic |
| 218 | C2×D4 | #685 | 1 | 3 | yes | — |
| 219 | C22×C4 | — | 10 | 6 | no | — |
| 220 | C22×C4 | C2×C4×D5 | 1 | 3 | yes | — |
| 221 | C22×C4 | C2×D20 | 1 | 3 | yes | — |
| 222 | C22×C4 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 223 | M16 | — | 8 | 3 | no | — |
| 224 | M16 | — | 40 | 3 | no | — |
| 225 | C2×C8 | — | 2 | 1 | no | — |
| 226 | C2×C8 | — | 2 | 3 | no | — |
| 227 | C2×C8 | — | 10 | 1 | no | — |
| 228 | C2×C8 | — | 10 | 3 | no | — |
| 229 | C4⋊C4 | — | 5 | 6 | no | — |
| 230 | C4⋊C4 | — | 5 | 2 | no | — |
| 231 | C4⋊C4 | — | 10 | 1 | no | — |
| 232 | C4⋊C4 | — | 10 | 3 | no | — |
| 233 | C4⋊C4 | — | 10 | 6 | no | — |
| 234 | C4⋊C4 | — | 10 | 6 | no | — |
| 235 | C4⋊C4 | — | 10 | 6 | no | — |
| 236 | C4⋊C4 | — | 10 | 6 | no | — |
| 237 | C4⋊C4 | #685 | 1 | 3 | yes | — |
| 238 | C4⋊C4 | #685 | 1 | 3 | yes | — |
| 239 | C22⋊C4 | — | 10 | 6 | no | — |
| 240 | C22⋊C4 | #685 | 1 | 6 | yes | — |
| 241 | C42 | — | 10 | 6 | no | — |
| 242 | C42 | C2×C4×D5 | 1 | 3 | yes | — |
| 243 | C42 | C2×D20 | 1 | 3 | yes | — |
| 244 | C42 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 245 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 246 | C2×C5 | — | 4 | 3 | no | — |
| 247 | C2×C5 | C22×#101 | 1 | 1 | yes | characteristic |
| 248 | C2×C5 | #2897 | 1 | 1 | yes | characteristic |
| 249 | C2×C5 | #2898 | 1 | 1 | yes | characteristic |
| 250 | C23 | C2×#685 | 1 | 3 | yes | — |
| 251 | Q8 | — | 20 | 3 | no | — |
| 252 | Q8 | — | 20 | 3 | no | — |
| 253 | Q8 | — | 20 | 1 | no | — |
| 254 | Q8 | — | 20 | 1 | no | — |
| 255 | Q8 | — | 40 | 6 | no | — |
| 256 | Q8 | #3956 | 1 | 4 | yes | — |
| 257 | D4 | — | 2 | 6 | no | — |
| 258 | C2×C4 | — | 2 | 3 | no | — |
| 259 | C2×C4 | — | 2 | 3 | no | — |
| 260 | C2×C4 | — | 2 | 3 | no | — |
| 261 | C2×C4 | — | 2 | 3 | no | — |
| 262 | C2×C4 | — | 2 | 6 | no | — |
| 263 | C2×C4 | — | 10 | 2 | no | — |
| 264 | C2×C4 | — | 10 | 6 | no | — |
| 265 | C2×C4 | — | 40 | 6 | no | — |
| 266 | C2×C4 | C2×#685 | 1 | 3 | yes | — |
| 267 | C2×C4 | C2×#685 | 1 | 3 | yes | — |
| 268 | C2×C4 | C2×#685 | 1 | 3 | yes | — |
| 269 | C2×C4 | C22×C4×D5 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | C22×D20 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 272 | C8 | — | 4 | 3 | no | — |
| 273 | C8 | — | 4 | 1 | no | — |
| 274 | C8 | — | 20 | 1 | no | — |
| 275 | C8 | — | 20 | 3 | no | — |
| 276 | C5 | #17777 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 277 | C22 | — | 2 | 3 | no | — |
| 278 | C22 | — | 2 | 3 | no | — |
| 279 | C22 | — | 2 | 3 | no | — |
| 280 | C22 | C22×#685 | 1 | 1 | yes | characteristic |
| 281 | C4 | — | 2 | 3 | no | — |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 2 | 3 | no | — |
| 284 | C4 | — | 20 | 2 | no | — |
| 285 | C4 | — | 20 | 6 | no | — |
| 286 | C4 | C2×#3956 | 1 | 6 | yes | — |
| 287 | C4 | #66085 | 1 | 1 | yes | characteristic |
| 288 | C4 | #66086 | 1 | 1 | yes | characteristic |
| 289 | C4 | #66170 | 1 | 1 | yes | characteristic |
| 290 | C4 | #66172 | 1 | 1 | yes | characteristic |
| 291 | C2 | — | 4 | 3 | no | — |
| 292 | C2 | #10614301 | 1 | 1 | yes | characteristic |
| 293 | C2 | C22×#3956 | 1 | 1 | yes | characteristic |
| 294 | C2 | #10615528 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta